Problem: Simplify the following expression: $ z = \dfrac{p + 3}{-6p + 2} - \dfrac{-7}{4} $
In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{4}{4}$ $ \dfrac{p + 3}{-6p + 2} \times \dfrac{4}{4} = \dfrac{4p + 12}{-24p + 8} $ Multiply the second expression by $\dfrac{-6p + 2}{-6p + 2}$ $ \dfrac{-7}{4} \times \dfrac{-6p + 2}{-6p + 2} = \dfrac{42p - 14}{-24p + 8} $ Therefore $ z = \dfrac{4p + 12}{-24p + 8} - \dfrac{42p - 14}{-24p + 8} $ Now the expressions have the same denominator we can simply subtract the numerators: $z = \dfrac{4p + 12 - (42p - 14) }{-24p + 8} $ Distribute the negative sign: $z = \dfrac{4p + 12 - 42p + 14}{-24p + 8}$ $z = \dfrac{-38p + 26}{-24p + 8}$ Simplify the expression by dividing the numerator and denominator by -2: $z = \dfrac{19p - 13}{12p - 4}$